Application of an internal tide generation model to baroclinic spring-neap cycles

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. C9, 3124, doi: /2001jc001177, 2002 Application of an internal tide generation model to baroclinic spring-neap cycles T. Gerkema Netherlands Institute for Sea Research, Texel, Netherlands Received 11 October 2001; revised 28 December 2001; accepted 10 January 2002; published 18 September [1] A numerical internal tide generation model is used to study the behavior of baroclinic spring-neap cycles in open basins and channels. The motivation for this study comes from an observation on tidal currents in Faeroe-Shetland Channel, which is briefly described; a surprisingly large baroclinic semidiurnal tidal signal is found in the lower part of the water column above the slope on the Shetland side, concurring with barotropic neap tides. The numerical model results indeed show that the baroclinic spring-neap cycle may have a phase shift with respect to the barotropic spring-neap cycle and that the phase of the baroclinic cycle may vary strongly within short distances. It is also shown that even small changes in the background conditions (e.g., stratification) can provoke a large phase shift in the baroclinic cycle; a possible connection to intermittency is discussed. INDEX TERMS: 4544 Oceanography: Physical: Internal and inertial waves; 4255 Oceanography: General: Numerical modeling; 4219 Oceanography: General: Continental shelf processes; KEYWORDS: internal tides, spring-neap cycles, intermittency Citation: Gerkema, T., Application of an internal tide generation model to baroclinic spring-neap cycles, J. Geophys. Res., 107(C9), 3124, doi: /2001jc001177, Introduction [2] In past decades it has been borne out by observations that internal tides can manifest themselves as beams in which the energy propagates in both the horizontal and the vertical direction [DeWitt et al., 1986; Pingree and New, 1989, 1991]. The angle of propagation a (with respect to the vertical) depends on the frequency s as tan 2 a ¼ N 2 s 2 s 2 f 2 ; where N is the buoyancy frequency and f is the Coriolis parameter [LeBlond and Mysak, 1978]. For N > f the right-hand side is a monotonically decreasing function of s. Hence a solar semidiurnal (S2) tidal beam propagates at a slightly steeper angle than does a lunar semidiurnal (M2) tidal beam; the beams thus follow different paths. This fact has not received much attention in the literature; it is the goal of this paper to explore some of its consequences. [3] First of all, it may explain why in some regions a baroclinic S2 signal can be found that is stronger than the baroclinic M2 signal even though the forcing (by the barotropic tide) of the latter is stronger; this was observed, for example, by Gould and McKee [1973] and Wang et al. [1991]. A further consequence is that the cophase lines of M2 do not coincide with those of S2, a fact that provides the key to the interpretation of the results presented in this paper. As will be derived below (section 3, equation (5)), Copyright 2002 by the American Geophysical Union /02/2001JC ð1þ the phase of the baroclinic spring-neap cycle depends on the phase difference between the M2 and S2 baroclinic signals. Therefore, because of the angle between the respective cophase lines, the moment of spring tides in the baroclinic signal will vary throughout the basin. So, in general, this signal will not be synchronous with the barotropic springneap cycle. [4] In recent acoustic Doppler current profiler (ADCP) observations from Faeroe-Shetland Channel ( presented in section 2) it was found that close to the bottom, a strong baroclinic semidiurnal signal coincides with barotropic neap tides (first quarter). The time series, however, covers only one spring-neap cycle and is therefore too short to warrant the conclusion that the strong baroclinic semidiurnal signal can indeed be interpreted as an occurrence of baroclinic spring tides (rather than alternative candidates, like an interaction between inertial and semidiurnal baroclinic signals). The goal of this paper therefore is to explore, by theoretical means, the possibility of such an interpretation. [5] The rest of this paper is organized as follows. In section 3 a simple internal tide generation model is presented, which will be used to analyze the baroclinic springneap cycle in a half-open basin (Bay of Biscay, section 4.1) and a channel-shaped basin (Faeroe-Shetland Channel, section 4.2). We emphasize that the goal of this paper is to shed light on the behavior of the baroclinic spring-neap cycle in a general and qualitative way, rather than to attempt to reproduce observations in much detail. In fact, it will be argued (section 5) that in certain regions it may be practically impossible to model the baroclinic spring-neap cycle faithfully because of the ever present noise in stratifica- 7-1

2 7-2 GERKEMA: BAROCLINIC SPRING-NEAP CYCLES Figure 1. The (a) along-slope and (b) cross-slope current (total signal, i.e., barotropic plus baroclinic) at 19 different depths; the scale on the left-hand side Figures 1a and 1b is indicative of the magnitude of the current (in cm s 1 ). tion and background currents to which the baroclinic springneap cycle may respond sensitively, giving the appearance of intermittency. 2. An Example From Faeroe-Shetland Channel [6] As part of the project Processes on the Continental Slope (PROCS) at Netherlands Institute for Sea Research (NIOZ), a cruise was made to Faeroe-Shetland Channel in spring 1999 [van Haren and van Raaphorst, 1999]. On the Shetland side slope (at position N, W; local depth 605 m) a long-range ADCP mooring was deployed, covering the vertical range from 15 m above the bottom to 94 m below the surface (divided in 128 bins of 4 m), sampling at a resolution of 300 s, during 13.2 days. Data from the upper 140 m proved largely worthless because of a lack of scatterers and have been removed. A presentation and discussion of the data was given by van Veldhoven [2000]; here we restrict ourselves to one salient feature in it. [7] Figure 1 shows the total (i.e., barotropic plus baroclinic) current speeds, decomposed into an along-slope component and an across-slope component ( positive values indicate a flow directed northeast and southeast, respectively). In both the semidiurnal tide is clearly present. Large semidiurnal oscillations occur over the whole column at the beginning and end of the period. This coincides with new moon (day 106) and full moon (day 120), as one would expect. However, in the lower 100 m of the water column, semidiurnal oscillations of the same order (or even larger) occur around first quarter (i.e., day 112), when one would expect neap tides (and hence a small signal). Being restricted to the lower part of the column, this large signal is evidently baroclinic. In other words, we must conclude that in these observations, relatively strong baroclinic tidal currents occur at a time when the barotropic tide is weak (barotropic neap tides). Phenomena like this have been described earlier elsewhere in the literature (see the discussion in section 5). In the rest of the paper this issue will be explored from a theoretical point of view. 3. Model Description [8] In the internal tide generation model used here we assume along-slope uniformity = 0); we can therefore introduce a stream function y (expressing the baroclinic cross-slope and vertical current speeds as u and w The linear hydrostatic equations then become [Gerkema, 3 @x þ N ¼ zn Q sin st dh ½H hx ðþš 2 dx : Here f is the Coriolis parameter, v is the transverse velocity component, and r is the density perturbation with respect to its local static value (multiplied by g/r *, for convenience, where g is the acceleration due to gravity and r * is the mean density). The right-hand side of equation (4) represents the forcing due to the barotropic tidal flow over the topography, in which Q is the amplitude of the cross-slope barotropic flux and s is the semidiurnal tidal frequency (either M2 or S2); the bottom lies at z = H + h(x), where H is the undisturbed ocean depth; and the upper surface (rigid lid) lies at z =0. [9] In reality the amplitude of the cross-slope barotropic flux Q would decrease oceanward on the scale of the (barotropic) Rossby radius of deformation, as well as ð2þ ð3þ ð4þ

3 GERKEMA: BAROCLINIC SPRING-NEAP CYCLES 7-3 coastward on the scale of the shelf width (see, e.g., the discussion by LeCann [1990, sect. 4.3]); however, since both scales are large compared to the width of the shelf break region, where the main generation of internal tides takes place, the assumption of a constant Q can be considered justifiable. [10] The equations are solved numerically for given stratification N(z) and topography h(x) (as well as given values for the constant parameters defined above: f, H, Q, and s). First, we transform the x, z domain to a rectangular shape, via the transformation h =1+2z/[H h(x)], where h denotes the new vertical coordinate. This allows the application of a pseudospectral method in the vertical, here involving Chebyshew polynomials. For the horizontal and time derivatives we use finite difference methods (centered differences in x and centered differences or third-order Adams-Bashforth in time; see [Durran, 1999]). At the outer ends of the domain, sponge layers are used to absorb the incoming waves. Separate runs are performed for the M2 case (s = s m ) and the S2 case (s = s s ). The fluid is initially at rest; the transients will have left the region of interest after a certain number of tidal periods (40, say). The variables, for instance, u m (the M2 cross-slope baroclinic component), then will have become periodic in time: u m ðt; x; zþ ¼ A m ðx; zþsin½s m t þ f m ðx; zþš; in which the amplitude A m and phase f m vary in space. An entirely similar result is obtained for the S2 run, yielding u s (replacing subscripts m by s). Afterward, the two results can be combined. A particularly convenient way to express the combined effect of M2 and S2 tides was described by Simpson et al. [1990]; thus we use the following identity: where u ¼ A m cosðs m t þ f m ÞþA s cosðs s t þ f s Þ ¼ At ðþcos½s m t þ fðþ t Š; A 2 ðþ¼a t 2 m þ A2 s þ 2A ma s cos cðþ; t tan fðþ¼ t A m sin f m þ A s sin½f m þ cðþ t Š A m cos f m þ A s cos½f m þ cðþ t Š ; cðþ¼ t ðs s s m Þt þ f s f m : Spring tides (i.e., A takes its maximum: A(t) =A m + A s ) occur when c(t) = 0, and neap tides (i.e., A takes its minimum: A(t)=jA m A s j) occur when c(t)=p. Hence the time at which baroclinic spring tides occur is given by t * ¼ f m f s s s s m ; which varies spatially via f m (x, z) and f s (x, z). 4. Model Results [11] In the first example we show results for conditions in the Bay of Biscay (in terms of stratification and topography). We start with this case because the internal tide beams travel only away from the continental slope and do not ð5þ return; hence the result is fairly transparent, and the beams can be easily identified. In the second case the conditions correspond to those in Faeroe-Shetland Channel. Here the beams reflect several times at the side slopes of the channel before they escape onto the shelves, and this gives rise to a much less transparent pattern. [12] Notice that an a priori estimate can be made of the difference in steepness between the M2 and S2 beams. Under the hydrostatic approximation the numerator on the right-hand side of equation (1) reduces to N 2 ; hence M2 and S2 beams traveling over the same horizontal distance travel over vertical distances (h m, h s ) that are related as h m ¼ s2 m f 2 1=2 h s s 2 s f 2 : For the Bay of Biscay this gives (see values given below) h m /h s = 0.92, and for the Faeroe-Shetland Channel (where f is larger) this gives h m /h s = Open Basin (Biscay) [13] The stratification N(z) is shown in Figure 2a; it involves a seasonal and permanent pycnocline and is representative of the summer conditions in the Bay of Biscay; the profile and topography are derived from Pingree and New [1991]. The barotropic tidal cross-slope flux is taken to be Q = 100 m 2 s 1, corresponding with current speeds of about 0.02 m s 1 in the deep ocean (comparable to values given by Pingree and New [1991, Table 1]). The M2 and S2 tidal frequencies are s m = s 1 and s s = s 1, and the Coriolis parameter at that latitude is f = s 1. For these frequencies the slope (see Figure 3) has two distinct supercritical regions, namely, one between 8.4 (8.8) and 25.6 (22.8) km and the other between 29.2 and 30.8 km (the values between parentheses refer to the values for the S2 beam, when different from those for the M2 beam). [14] In the runs, 64 Chebyshew polynomials were used, and the steps in x and t (centered differences) were 400 m and 120 s. The results of the separate runs for M2 and S2 can be combined afterward, following the procedure outlined in section 3. In accordance with data from Table 1 by Pingree and New [1991] we chose the M2 forcing to be 1.6 times stronger than the S2 forcing. [15] Figure 3 shows the results: the total amplitude A m + A s (the maximum of amplitude A(t)), the relative strength of M2, and the moment at which spring tides occur in the baroclinic signal. All these quantities vary strongly spatially. In Figure 3a one can identify two beams that emanate from the upper parts of the slope; one of them travels directly downward into the basin, and the other one travels first upward to the surface. The former is the beam Pingree and New [1991] focused on; their mooring 116, which was located in the region where the reflection at the bottom takes place, corresponds to x = 70 km in Figure 3. At this position we find a maximum current speed (near the bottom) of about 0.14 m s 1, which agrees fairly well with the sum of the observed baroclinic M2 (10.4 cm s 1 ) and S2 (1.7 cm s 1 ) contributions [Pingree and New, 1991, Table 1]. At higher positions at x = 70 km in Figure 3a the total amplitude A m + A s first decreases but then takes again larger

4 7-4 GERKEMA: BAROCLINIC SPRING-NEAP CYCLES Figure 2. Stratification profiles used in the numerical calculations: (a) the profile in the Bay of Biscay [after Pingree and New, 1991, Table 1] and (b) a profile (solid line) derived from sections in Faeroe- Shetland Channel [after van Haren and van Raaphorst, 1999]; the dashed line shows a hypothetically modified profile. Figure 3. A numerical calculation of the combined effect of M2 and S2 internal tidal beams in the Bay of Biscay: (a) A m + A s, i.e., the amplitude of the baroclinic cross-slope current speed at spring tides, (b) the the relative importance of the components, (A m A s )/(A m + A s ), and (c) the spatial distribution of the time (in days) at which spring tides occur in the baroclinic signal. Figure 3c is cyclic: white (day 0) denotes the same phase as black (day 14.84).

5 GERKEMA: BAROCLINIC SPRING-NEAP CYCLES 7-5 Figure 4. A numerical calculation of the combined effect of M2 and S2 internal tidal beams in Faeroe- Shetland Channel: (a) A m + A s, i.e., the amplitude of the baroclinic cross-slope current speed at spring tides, (b) the the relative importance of the components, (A m A s )/(A m + A s ), and (c) the spatial distribution of the time (in days) at which spring tides occur in the baroclinic signal. Figure 4c is cyclic: white (day 0) denotes the same phase as black (day 14.84). values at depths between 2000 and 500 m; this is in qualitative agreement with the observations. [16] Figure 3b shows the relative importance of the two baroclinic semidiurnal components; values can range from 1 (complete S2 dominance) to +1 (complete M2 dominance). Overall, the M2 component is stronger (not surprisingly, in view of the fact that the M2 forcing is 1.6 times stronger). However, in some regions the S2 signal becomes equally (or even more) important, for instance, near the bottom around 60 km, indicating that the S2 beam reflects before the M2 beam does (i.e., more to the left); this is in accordance with the fact (mentioned in section 1) that the S2 beam travels at a steeper angle. [17] Finally, Figure 3c shows the moment at which spring tides occur in the baroclinic signal; this moment represents the shift with respect to the barotropic spring tides, which occur at day 0 (white) and, one cycle later, at day (black). Particularly interesting is the region near x = 68 km at about 3 km depth, where the phase of the spring-neap cycle varies strongly in the vertical; however, this would probably not be very noticeable in a real signal since in this region the total amplitude is small (see Figure 3a). Near x = 85 km, where the total amplitude takes fairly significant values throughout the water column, we find phase differences of some 4 days over the vertical. In the main area of reflection (near the bottom around x = 70 km) the baroclinic spring-neap cycle lags the barotropic cycle by about 2 days Channel-Shaped Basin (Faeroe-Shetland Channel) [18] For the stratification we use the cross-channel average of observed vertical profiles from Faeroe-Shetland Channel [van Haren and van Raaphorst, 1999] (see Figure 2b (solid line)); in the upper and lower part of the column, N has simply been taken as constant since the data show little variation. The Coriolis parameter is now f = s 1. The (total) barotropic tidal cross-slope flux is taken to be Q = 60 m 2 s 1, and the ratio of M2 and S2 forcing strength is 2.0 (these values are based on the observed cross-slope component, shown in Figure 1, from which the depthaveraged signal was derived and analyzed). The uniformity in the along-channel direction, assumed in section 3, means that we ignore possible effects of an along-channel propagating internal tide generated at the Wyville-Thomson Ridge (observations by Sherwin [1991] pointed to significant generation at the ridge). [19] In the calculations, 50 Chebyshew polynomials were used in the vertical, and horizontal and time steps were 500 m and 70 s (here a third-order Adams-Bashforth scheme was used for the time integration). A simple Rayleigh friction term was added, with a damping timescale of 15 periods (i.e., about 1 week); this weak frictional term suffices to mitigate the growth near critical regions at the slope, where the signal would otherwise grow linearly with time (in accordance with the theoretical analysis by Dauxois and Young [1999]). [20] The results from the numerical calculation (obtained after 100 tidal periods, when the transients have left the domain) are shown in Figure 4. Relatively strong currents are found near the upper part of the slope (on both sides of the channel), between 300 and 600 m depth (see Figure 4)a; this is in qualitative agreement with observations by van Raaphorst et al. [2001]. In particular, at x = 215 km

6 7-6 GERKEMA: BAROCLINIC SPRING-NEAP CYCLES Figure 5. The shift in the baroclinic spring-neap cycle (in days) due to the change of stratification shown in Figure 2b (solid line versus dashed line). The plot is cyclic: white and black both denote a zero shift (or nearly so); medium shading indicates a shift of about 1 week. (corresponding to the observational site; section 2) the largest values, between 0.2 and 0.3 m s 1, occur (only) within the lowest 100 m of the water column. This is in agreement with the observations shown in Figure 1, where between days 112 and 114 (the reason for selecting these days is given below when we discuss the phase shifts in the baroclinic spring-neap cycle) the amplitude in the crossslope component reaches maximum values of about m s 1 in the lowest part of the water column. [21] The paths along which the internal tides propagate can be identified but form a somewhat blurred pattern because the beams traverse the channel several times before they escape onto the shelves, thus giving rise to a complicated web. In these (and other similar) model runs, no indications are found that closed paths exist, i.e., attractors in the sense of Maas and Lam [1995] and Maas et al. [1997]. A necessary condition for the occurrence of these attractors is that supercritical regions are present at both sides of the basin; this condition is satisfied here since four such regions exist: on one side between 87.0 and 98.5 km and around 104 km for M2 (for S2: between 87.5 and 92.0 km and between 94.5 and 98.5 km) and on the other side between and km and between and km (for S2: between and km and between and km). Nevertheless, no attractors were found; this can be ascribed to the fact that the basin is not entirely closed, so that the beams can (and in fact do) escape onto the shelves. It should be noted, however, that the structure of the web depends on the wave frequency and that for internal waves of a different frequency (i.e., not semidiurnal), but for the same stratification and topography, attractors may exist. [22] Figure 4b shows that M2 is dominant in the baroclinic signal, except at a few distinct regions. Figure 4c shows the day at which spring tides occur in the baroclinic signal; it varies strongly spatially. However, the regions where the variation is strongest more or less coincide with those in which the amplitude (Figure 4a) is small, the same phenomenon as we saw in section 4.1. Nevertheless, a significant variation is present; in particular, on the slope on the right-hand side in Figure 4c (i.e., the Shetland side), at x = 215 km (the observational site), we find that in the lowest 200 m baroclinic spring tides lag the barotropic spring tides by 4 7 days. This is in agreement with the observations shown in Figure 1, where in the lowest part of the water column strong semidiurnal baroclinic signals occur some 6 days after barotropic spring tides. (However, no excessive importance should be attached to this agreement, in view of the points discussed in section 5.) [23] As we saw above (equation (5)), the moment at which baroclinic spring tides occur depends spatially on the local difference in phase between M2 and S2. These phases, in turn, depend on the trajectories that the beams follow, and hence on background conditions like stratification N. Thus one would expect that even a minor variation (in time) in, for example, N, which changes the paths of the beams, may at some positions give rise to a drastic shift in the baroclinic spring-neap cycle. To verify this idea, we carried out a calculation identical to the one that led to Figure 4, except that N was slightly different, being now given by the dashed line in Figure 2b. The outcome looks very much like Figure 4 (and is therefore not shown), but interesting details are revealed if we plot the local shift in the baroclinic spring-neap cycle

7 GERKEMA: BAROCLINIC SPRING-NEAP CYCLES 7-7 (see Figure 5). At most positions the (small) change in stratification gives no significant shift in the cycle, but at some positions a shift as large as a week is found (notice that this happens in the regions where the spatial variation shown in Figure 4c is strong). Clearly, at such positions one would never expect to find a consistent spring-neap cycle since in nature, small variations in background conditions (noise) are always present. 5. Conclusion and Discussion [24] With regard to the behavior of baroclinic springneap cycles, three general conclusions can be drawn from the numerical results presented in section 4. First, the phase of this cycle may vary strongly spatially; for example, one may find baroclinic spring tides at some position and simultaneously neap tides at a nearby position. Second, precisely in those regions where such a strong spatial variability exists, the phase of the baroclinic spring-neap cycle becomes sensitive to small variations in background conditions, like the stratification N. Third, this sensitivity to background conditions, which are always only imperfectly known from observations, makes it unlikely that the baroclinic spring-neap cycle can be modeled faithfully in such regions. [25] These conclusions suggest a link with the occurrence of intermittency. In past decades, abundant evidence has been gathered of intermittent behavior in internal-tide signals; they come and go [Wunsch, 1975]. The classic example is the observation by Magaard and McKee [1973]. Importantly, as their current meter data show, the intermittency manifests itself not in the semidiurnal signal as such (this signal is persistent and identifiable throughout the time series) but rather in its envelope (i.e., at a timescale of several days to a week), in other words, in the absence of a persistent spring-neap cycle. Similarly, in current measurements on the northwest African continental slope [Huthnance and Baines, 1982], no persistent spring-neap cycles can be seen in the current data, although both M2 and S2 are prominently present in the spectra. At a more southern position on the same slope [Schott, 1977, Figure 6], current measurements at deep positions follow the barotropic spring-neap cycle, while at higher positions the cycle (if present at all) seems shifted in time. Siedler and Paul [1991] found from current measurements in the eastern North Atlantic that the envelope of the semidiurnal baroclinic signal may have timescales about twice as short as a spring-neap cycle. On the Malin Shelf, horizontal incongruities are found [Sherwin, 1988]; no clear spring-neap cycles are observed near the shelf break, in contrast to a position farther on-shelf. [26] So, these observations have in common that the intermittency manifests itself predominantly as a lack of a persistent spring-neap cycle. Intermittency has been variously attributed to variability in the stratification near the shelf break [Huthnance and Baines, 1982; Baines, 1986] or to irregularities in the cross-slope currents due to wind and upwelling [Sandstrom, 1991]. Either way, the background conditions, and hence the paths of the internal tide beams, will vary, perhaps only slightly, but this may cause the phase of the baroclinic spring-neap cycle to wander all the time. The results of this paper suggest that this phenomenon can in principle be described by the simple linear model of section 3, if the prescribed quantities N (stratification) and Q (barotropic flux) are subject to noise. [27] Acknowledgments. This work was carried out within the NIOZ project PROCS. The author is grateful to H. van Haren and L. R. M. Maas for valuable comments on an earlier version of this paper. The observational material (Figure 1) was generously provided by A. K. van Veldhoven and H. van Haren. This is NIOZ publication References Baines, P. G., Internal tides, internal waves, and near-inertial motions, in Baroclinic Processes on Continental Shelves, Coastal Estuarine Sci., vol. 3, edited by C. N. K. Mooers, pp , AGU, Washington, D. C., Dauxois, T., and W. R. Young, Near-critical reflection of internal waves, J. Fluid Mech., 390, , DeWitt, L. M., M. D. Levine, C. A. Paulson, and W. V. Burt, Semidiurnal internal tide in JASIN: Observations and simulation, J. Geophys. 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Gerkema, Netherlands Institute for Sea Research, PO Box 59, 1790 AB Den Burg, Texel, Netherlands. (gerk@nioz.nl)